6 POSITIVE LINEAR FORMS, POSITIVE HUBERT FORMS 353
When A has a unit element and is a Banach algebra with involution, not
only does the theory of positive Hilbert forms reduce to that of positive linear
forms, and the condition (N) is automatically satisfied, but also the condition
(U) is satisfied. Precisely,
(15.6.11) Let A. be a Banach algebra with involution, having a unit element
e ^ 0. Let f be a positive linear form on A. Then
(i) fis continuous and \\f\\ = f(e).
(ii) \f(y*xy)\ £ \\x\\f (y*y).
We shall use the following lemma:
(22.214.171.124) If xe A is self-adjoint and \\x\\ < 1, then there exists a self-
adjoint y e A such that y2 = e + x.
If t is real and \t\<\9 Taylor's formula (8.14.3) gives
2-4---2n ol (1 + s)"2'
This leads immediately to the estimate
and therefore we have
(126.96.36.199) (1 + 01/2 = 1 + 4*
the series on the right being absolutely convergent for \t \ < 1 (9.1 .2). It follows
that in the Banach algebra A the series
converges absolutely for \\x\\ < 1, and its sum y is self-adjoint when x is self-
adjoint. Furthermore, since the square of the power series on the right-hand
side of (188.8.131.52) is 1 + t, it follows that y2 = e 4- x, by virtue of (5.5.3).
This lemma shows that, if xe A is self-adjoint and ||jc|| < 1, there exists
y e A such that y*y = e — x, so that/(e — x) ^ 0, or equivalently/(;*;) £f(e).
If now ze A is such that ||z|| < 1, then also \\z*z\\ < 1 and therefore, usingrate (15.5.5). This is equivalent to the following condition on g: